Using Product and Quotient Properties for Exponents

5.1.2 Using Product and Quotient Properties for Exponents

5.1.2 • Using Product and Quotient Properties for Exponents

Activity

Product Property for Exponents

If $a$ is a real number and $m$ and $n$ are integers, then

$a^{m} \cdot a^{n} = a^{m + n}$

To multiply with like bases, add the exponents.

Complete problems 1 - 6 using the product property for exponents.

1. $d^{2} \cdot d^{3}$

Compare your answer:

$d^{2} \cdot d^{3} = d^{2 + 3} = d^{5}$

2. $c^{4} \cdot c^{6}$

Compare your answer:

$c^{4} \cdot c^{6} = c^{4 + 6} = c^{10}$

3. $k^{3} \cdot k^{5}$

Compare your answer:

$k^{3} \cdot k^{5} = k^{3 + 5} = k^{8}$

4. $(- v)^{2} \cdot (- v)^{1}$

Compare your answer:

$(- v)^{2} \cdot (- v)^{1} = (- v)^{2 + 1} = (- v)^{3}$

5. $(m)^{4} \cdot (m)^{4}$

Compare your answer:

$(m)^{4} \cdot (m)^{4} = m^{4 + 4} = m^{8}$

6. $(- p)^{6} \cdot (- p)^{8}$

Compare your answer:

$(- p)^{6} \cdot (- p)^{8} = (- p)^{6 + 8} = (- p)^{14}$

Product Quotient for Exponents

If $a$ is a real number, $a \neq 0$ , and $m$ and $n$ are integers, then

$\frac{a^{m}}{a^{n}} = a^{m - n}$ , $m > n$ and $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ , $n > m$

To divide with like bases, subtract the exponents.

Complete problems 7 - 12 using the quotient property for exponents.

7. $\frac{y^{5}}{y^{3}}$

Compare your answer:

$\frac{y^{5}}{y^{3}} = (y)^{5 - 3} = (7)^{2}$

8. $\frac{r^{4}}{r}$

Compare your answer:

$\frac{r^{4}}{r} = (r)^{4 - 1} = (r)^{3}$

9. $\frac{q^{9}}{q^{2}}$

Compare your answer:

$\frac{q^{9}}{q^{2}} = (q)^{9 - 2} = (q)^{7}$

10. $\frac{(- u)^{5}}{(- u)^{2}}$

Compare your answer:

$\frac{(- u)^{5}}{(- u)^{2}} = (- u)^{5 - 2} = (- u)^{3}$

11. $\frac{(j)^{6}}{(j)^{5}}$

Compare your answer:

$\frac{(j)^{6}}{(j)^{5}} = (j)^{6 - 5} = (j)$

12. $\frac{(- z)^{8}}{(- z)^{6}}$

Compare your answer:

$\frac{(- z)^{8}}{(- z)^{6}} = (- z)^{8 - 6} = (- z)^{2}$

Self Check

Which of the following equations is correct?

$\frac{(-u)^8}{(-u)^2}=2(-u)^4$
$\frac{y^6}{y^3}=y^2$
$x^2 \cdot x^4=x^8$
$y^5 \cdot y^3=y^8$

Additional Resources

Using Product and Quotient Properties for Exponents

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property.

What does $x^{2} \cdot x^{3}$ mean?

An equation shows two x multiplied (2 factors) and three x multiplied (3 factors), separated by a dot, combining to make five x multiplied (5 factors).

Notice that 5 is the sum of the exponents, 2 and 3. We see $x^{2} \cdot x^{3}$ is $x^{2 + 3}$ or $x^{5}$ .

The base stayed the same, and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If $a$ is a real number and $m$ and $n$ are integers, then

$a^{m} \cdot a^{n} = a^{m + n}$

To multiply with like bases, add the exponents.

Example 1

$4^{3} \cdot 4^{4} = 4^{3 + 4} = 4^{7}$

Example 2

$12^{2} \cdot 12^{6} = 12^{2 + 6} = 12^{8}$

Example 3

$y^{7} \cdot 4^{8} = y^{7 + 8} = y^{15}$

Example 4

$(2^{b})^{10} \cdot (2 b)^{3} = (2 b)^{10 + 3} = (2 b)^{13}$

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

1. Consider $\frac{x^{5}}{x^{2}}$

Step 1 - What does it mean?

$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$

Step 2 - Use the Equivalent Fractions Property.

$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$

Step 3 - Simplify.

$x^{3}$

2. Consider $\frac{x^{2}}{x^{3}}$

Step 1 - What does it mean?

$\frac{x \cdot x}{x \cdot x \cdot x}$

Step 2 - Use the Equivalent Fractions Property.

$\frac{x \cdot x \cdot 1}{x \cdot x \cdot x}$

Step 3 - Simplify.

$\frac{1}{x}$

Notice that in each case, the bases were the same and we subtracted exponents. We see $\frac{x^{5}}{x^{2}}$ is $x^{5 - 2}$ or $x^{3}$ . We see $\frac{x^{2}}{x^{3}}$ is $\frac{1}{x}$ . When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, so we need a 1 in the numerator. $\frac{x}{x} = 1$ . This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If $a$ is a real number, $a \neq 0$ , and $m$ and $n$ are integers, then

$\frac{a^{m}}{a^{n}} = a^{m - n}$ , $m > n$ and $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ , $n > m$

Example 5

$7^{5} \div 7^{3} = 7^{5 - 3} = 7^{2}$

Example 6

$k^{9} \div k^{2} = k^{9 - 2} = k^{7}$

Example 7

$\frac{j^{10}}{j^{7}} = j^{10 - 7} = k^{3}$

Try it

Using Product and Quotient Properties for Exponents

Simplify each expression.

$12^{2} \cdot 12^{5}$
$20^{14} \div 20^{4}$
$h^{4} \cdot h^{7}$
$(- s)^{2} \cdot (- s)^{2}$
$\frac{b^{7}}{b^{5}}$
$(- y)^{6} (- y)^{3}$
$\frac{x^{3}}{x^{5}}$

Compare your answers:

Here is how to simplify the expressions:

$12^{2} \cdot 12^{5} = 12^{7}$
$20^{14} \div 20^{4} = 20^{10}$
$h^{4} \cdot h^{7} = h^{11}$
$(- s)^{2} (- s)^{2} = (- s)^{4}$
$\frac{b^{7}}{b^{5}} = b^{2}$
$(- y)^{6} (- y)^{3} = (- y)^{9}$
$\frac{x^{3}}{x^{5}} = \frac{1}{x^{2}}$

Video: Understanding Product and Quotient Properties for Exponents

Watch the following video to learn more about applying the Product and Quotient Properties of Exponents.

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